In known manner, a fluid dispenser pump generally includes a pump body in which a piston is mounted to slide between a rest position and a dispensing position. The pump body is cylindrical and the end positions of the piston, namely the rest position and the dispensing position, are defined by various means, e.g. abutment means. When the pump is actuated, the pump chamber is filled with fluid, and the leaktightness constraints for the piston while it is moving inside the pump body are very stringent to avoid any risk of leakage. Therefore, the pump must exert non-negligible radial stresses on the piston while it is moving towards its dispensing position. Unfortunately, most pumps are used at irregular intervals, and the pump spends most of its time in the rest position. The high radial stresses that are exerted on the piston, also when it is in the rest position, can be disadvantageous in that ultimately they can damage the sealing lips of the piston or can deform them, thereby diminishing their leaktightness, which can cause the pump to malfunction. Furthermore, certain pumps use a second piston that is also mounted to move in the pump body between a respective rest position and a respective dispensing position. The second piston is generally mounted to slide in a small-diameter portion of the pump body. In that type of pump, the second piston forms or is connected in some way to the inlet valve and/or the outlet valve of pump chamber. Similarly to what is described above, the pump body is cylindrical and the second piston is mounted to slide in leaktight manner in said cylindrical portion of the pump body between its rest position and its dispensing position. For the second piston, there is a risk that, when it returns from its dispensing position to its rest position, i.e. generally during the stage in which the fluid contained in the pump chamber is expelled, the suction generated by said expulsion inside the pump chamber might cause the inlet valve of the pump to open early, or might cause the outlet valve of the pump chamber to close early, when valve opening and closure is controlled by said second piston. To solve that problem, it is possible to make provision to increase the radial stresses exerted by the pump body on the second piston. However, that requires a larger actuating force to actuate the pump. Such malfunctioning of the valves of the pump chamber can result in non-uniform quantities or “doses” of fluid being dispensed because the times at which said valves open and close are no longer accurately predeterminable and reproducible each time the pump is actuated.